CN113221260B - Vibration control method based on bifurcation and chaotic analysis - Google Patents

Vibration control method based on bifurcation and chaotic analysis Download PDF

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CN113221260B
CN113221260B CN202011464594.3A CN202011464594A CN113221260B CN 113221260 B CN113221260 B CN 113221260B CN 202011464594 A CN202011464594 A CN 202011464594A CN 113221260 B CN113221260 B CN 113221260B
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CN113221260A (en
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吴石
刘震
张轩瑞
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Harbin University of Science and Technology
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Abstract

The invention relates to a method for controlling a workpiece vibration system in the milling process of a thin-wall workpiece by bifurcation and chaos analysis, which comprises the following steps: firstly, establishing a nonlinear dynamics equation of a rectangular thin-wall piece under the action of thermal coupling in the milling process; then revealing the condition of generating Smale horseshoe chaos by a nonlinear vibration system through a Melnikov function; meanwhile, according to the influence rule of bifurcation and chaos analysis methods on the nonlinear dynamic behavior of the system under the changes of load, temperature, technological parameters and the like in the thin-wall part processing process, finally, a thin-wall part vibration control method optimized through the processing parameters is established. According to the invention, from the established dynamic equation of the workpiece vibration system in the machining process, the bifurcation and chaos analysis method is applied to the influence rule of the workpiece vibration system under the changes of load, temperature, technological parameters and the like in the milling process. The method for analyzing and controlling vibration by utilizing chaos and bifurcation can scientifically select processing parameters, thereby effectively improving processing quality.

Description

Vibration control method based on bifurcation and chaotic analysis
Technical Field
The invention relates to the field of vibration control of thin-wall parts, in particular to a vibration control method based on bifurcation and chaos analysis.
Background
The thin-wall part is widely applied to a plurality of important fields such as aerospace, energy, traffic and the like, and milling vibration of the thin-wall part has rich nonlinear behavior characteristics in the machining process. In order to control milling vibration in the machining process and improve the machining quality and the machining precision of the surface of the thin-wall part, it is very necessary to research bifurcation and chaos of vibration of the thin-wall part with small deflection under the action of a thermal coupling field and a force coupling field.
The external excitation has a great influence on the nonlinear vibration of the thin-walled workpiece. Since the 21 st century, yeh, Y. -L. and the like studied the law of influence of thermal coupling on nonlinear vibration of a large-deflection sheet; abdulkrim S and the like find that with the increase of excitation intensity, when the transverse deflection of the thin-wall part reaches 50% of the thickness of the thin-wall part, the system generates bifurcation phenomenon and the like; the Awrejcewicz J and the like carry out chaos analysis of nonlinear vibration on thin shells under different degrees of freedom based on Galerkin method; he x, et al analyze nonlinear vibrations of the thin-walled plate under thermal load based on total energy weighted margin; zhang W et al studied the nonlinear vibration variation of the rotating thin-walled disc to find that damping and external excitation have the greatest effect on the system, and when the damping reaches a certain value, the system tends to be in a stable state; in addition, domestic plateau texts and the like analyze the chaotic vibration phenomenon of the ferromagnetic beam plate in a periodic time-varying magnetic field.
According to the invention, a nonlinear dynamics equation of a rectangular thin-wall part under the action of thermal coupling is established, then the condition of the system for generating Smale horseshoe chaotic vibration is revealed, the influence rule of load, temperature, technological parameters and the like on the nonlinear dynamics behavior of the system in the processing process of the thin-wall part is analyzed, and finally a thin-wall part vibration control method optimized by the processing parameters is established.
Disclosure of Invention
The invention aims to provide a vibration control method based on bifurcation and chaos theory.
In order to solve the problems, the invention adopts the following technical scheme: firstly, establishing a milling process dynamics model, which is characterized in that: the milling force model is divided into 2 parts:
the first part builds a dynamic milling force model, comprising the following steps:
1) A physical model of milling dynamics is built, which consists of two degrees of freedom elastic damping systems perpendicular to each other:
Figure SMS_1
Figure SMS_2
wherein a is p -milling depth; Δx, Δy—regenerative chatter, by the difference between the vibration displacements of the current tooth and the previous milling tooth, t—the rotation period of each tooth, t=2pi/(nω), n—the number of teeth of the milling cutter; omega-spindle angular velocity; a, a ij (t) -time-varying milling force coefficient, i, j=1, 2, a when the milling cutter tooth is between the cut-in angle and the cut-out angle ij (t) may be expressed as an average of the cutter tooth position angles.
For simplicity, as shown in fig. 2, the milling system is a single degree of freedom system, w (t) is the dynamic displacement of the system in the modal direction, and the included angle between w and the x axis is alpha. F (F) x,y Is the position angle phi of the cutter tooth j j Time F x ，F y The resulting milling forces, i.e. the resulting dynamic milling forces, are projected in the modal direction, the values of which can be expressed as:
Figure SMS_3
considering the case where w (t) is a constant amplitude harmonic, it is possible to obtain:
Figure SMS_4
wherein a is 0 ＝a 11 cos 2 α+a 12 sinαcosα-a 21 sinαcosα-a 22 sin 2 Alpha. When the system reaches an arbitrary time lag critical steady-state value, i.e. critical state between steady and unsteady
Figure SMS_5
The second portion models the thin wall panel milling force process.
Balance equation for thin-walled rectangular plates under milling forces and temperature variations caused by milling forces during milling:
Figure SMS_6
wherein:
w is the dynamic displacement of the system in the mode direction, v is poisson's ratio, ρ is the density of the rectangular thin-wall plate, h is the thickness of the rectangular thin-wall plate, M T Is the heat moment, N T Is the heat power, the heat power is the heat power,
Figure SMS_7
as a stress function->
Figure SMS_8
For flexural rigidity, E is modulus of elasticity
Figure SMS_9
The expression of the thermal force and thermal moment is:
Figure SMS_10
alpha in the formula 0 Is the linear expansion coefficient of the material; t (x, y, z) is a temperature function generated by cutting forces during milling; because the four sides of the rectangular thin plate are immovable simple supports, a and b are the side lengths of the rectangular plate, and the boundary conditions are as follows:
Figure SMS_11
the heat generated by the milling force, the dynamic displacement of the system in the modal direction, are assumed to be respectively:
Figure SMS_12
wherein: t (T) 0 Is a temperature constant; a (t) is a functional expression with respect to time t.
The second formula (7), the second formula (9) and the second formula (5) can be obtained:
Figure SMS_13
according to the undetermined coefficient method, setting a stress function as follows:
Figure SMS_14
bringing it into the above, comparing the coefficients at both sides to obtain the stress function coefficient
Figure SMS_15
The first formula in formula (5) is obtainable using the Galerkin principle:
Figure SMS_16
substituting the formula (7), the formula (9) and the formula (13) into the above formula, the calculation can be obtained:
Figure SMS_17
wherein:
Figure SMS_18
Figure SMS_19
Figure SMS_20
Figure SMS_21
from the expression c 3 >0，c 4 >0, using dimensionless method, let:
Figure SMS_22
then equation (14) ultimately reduces to:
Figure SMS_23
order the
Figure SMS_24
μ=εδ+fcos (ωτ) then the system equation (17) equivalent system is:
Figure SMS_25
equation (18) is an undisturbed Hamilton system when ε=0, i.e
Figure SMS_26
It has three singular points: two saddle points (+ -1, 0) and a center (0, 0).
The Hamilton quantity of system formula (19) is:
Figure SMS_27
the trajectory into saddle point (0, 0) should be h when t tends to plus or minus infinity 0 =0, i.e.:
Figure SMS_28
two sink tracks with respect to time can be obtained by the above method:
Figure SMS_29
the above formula is expressed by integration and by an elliptic function:
Figure SMS_30
wherein: dn, sn and cn are Jacobi elliptic functions, q is an elliptic modulus. Closed rail period:
Figure SMS_31
wherein: k (q) is a first type of elliptic integral; t (T) q The elliptical mode q is monotonically increasing. />
Figure SMS_32
And->
Figure SMS_33
And->
Figure SMS_34
The Melnikov function for the cognate track is:
Figure SMS_35
calculated by
Figure SMS_36
When (when)
Figure SMS_37
When established, the Melnikov function M (t 0 ) There is a simple zero point. Therefore, under the above conditions, the system has chaos in the sense of a Smale horseshoe.
The beneficial effects of the invention are as follows:
the invention relates to a processing parameter optimization method based on bifurcation and chaotic analysis, which comprises the steps of firstly establishing a dynamic milling force model in a modal direction, establishing a rectangular thin-wall part milling dynamics equation under the action of milling heat according to plate shell elastic mechanics, and judging chaotic vibration conditions under Smale horseshoe transformation generated by a nonlinear dynamics system based on a Melnikov function method. And (3) solving a bifurcation diagram, a displacement waveform diagram, a phase plane track diagram and a Poincare diagram of the nonlinear system of the thin-wall part under the action of milling force, quantitatively analyzing the influence rules of cutting depth, cutting temperature, thickness of the thin-wall part, spindle rotating speed and the like on the vibration of the nonlinear system, and stably predicting the nonlinear system of the thin-wall part (such as spindle rotating speed, cutting depth and the like). The accuracy is verified through the comparison of actual machining and theory, the parameter optimization can be carried out before machining, and guidance is provided for machining.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions of the prior art, the drawings that are needed in the embodiments will be briefly described below, it being obvious that the drawings in the following description are only some embodiments of the present invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a physical model of milling dynamics;
FIG. 2 is a simplified vibration mode model of a single degree of freedom milling system;
FIG. 3 is a diagram of a bifurcation of system vibration with spindle variation;
FIG. 4 is a diagram of the bifurcation of system vibration as a function of milling depth;
FIG. 5 is a graph of vibration bifurcation with temperature change of the thin-walled plate surface;
fig. 6 is a diagram showing the bifurcation of system vibration according to the thickness of the plate.
FIG. 7 is a schematic diagram of the experimental test system of the present invention.
FIG. 8 is a graph showing the relationship between spindle rotation speed and cutting depth and vibration acceleration according to the present invention, (a) a first experimental graph, and (b) a second experimental graph;
FIG. 9 is a graph showing the relationship between spindle rotation speed and thin-walled member thickness and vibration acceleration established in the experiment of the present invention, (a) a first experimental graph, and (b) a second experimental graph;
Detailed Description
The first part builds a dynamic milling force model, comprising the following steps:
1) A physical model of milling dynamics is built, which consists of two degrees of freedom elastic damping systems perpendicular to each other:
Figure SMS_38
Figure SMS_39
wherein a is p -milling depth; Δx, Δy—regenerative chatter, by the difference between the vibration displacements of the current tooth and the previous milling tooth, t—the rotation period of each tooth, t=2pi/(nω), n—the number of teeth of the milling cutter; omega-spindle angular velocity; a, a ij (t) -time-varying milling force coefficient, i, j=1, 2, a when the milling cutter tooth is between the cut-in angle and the cut-out angle ij (t) may be expressed as an average of the cutter tooth position angles.
For simplicity, as shown in fig. 2, the milling system is provided as a single degree of freedom system,w (t) is the dynamic displacement of the system in the modal direction, and the included angle between w and the x axis is alpha. F (F) x,y Is the position angle phi of the cutter tooth j j Time F x ，F y The resulting milling forces, i.e. the resulting dynamic milling forces, are projected in the modal direction, the values of which can be expressed as:
Figure SMS_40
considering the case where w (t) is a constant amplitude harmonic, it is possible to obtain:
Figure SMS_41
wherein a is 0 ＝a 11 cos 2 α+a 12 sinαcosα-a 21 sinαcosα-a 22 sin 2 Alpha. When the system reaches an arbitrary time lag critical steady-state value, i.e. critical state between steady and unsteady
Figure SMS_42
The second portion models the thin wall panel milling force process.
Balance equation for thin-walled rectangular plates under milling forces and temperature variations caused by milling forces during milling:
Figure SMS_43
wherein:
w is the dynamic displacement of the system in the mode direction, v is poisson's ratio, ρ is the density of the rectangular thin-wall plate, h is the thickness of the rectangular thin-wall plate, M T Is the heat moment, N T Is the heat power, the heat power is the heat power,
Figure SMS_44
as a stress function->
Figure SMS_45
For flexural rigidity, E is modulus of elasticity
Figure SMS_46
The expression of the thermal force and thermal moment is:
Figure SMS_47
alpha in the formula 0 Is the linear expansion coefficient of the material; t (x, y, z) is a temperature function generated by cutting forces during milling; because the four sides of the rectangular thin plate are immovable simple supports, a and b are the side lengths of the rectangular plate, and the boundary conditions are as follows:
Figure SMS_48
the heat generated by the milling force, the dynamic displacement of the system in the modal direction, are assumed to be respectively:
Figure SMS_49
wherein: t (T) 0 Is a temperature constant; a (t) is a functional expression with respect to time t.
The second formula (7), the second formula (9) and the second formula (5) can be obtained:
Figure SMS_50
according to the undetermined coefficient method, setting a stress function as follows:
Figure SMS_51
bringing it into the above, comparing the coefficients at both sides to obtain the stress function coefficient
Figure SMS_52
The first formula in formula (5) is obtainable using the Galerkin principle:
Figure SMS_53
substituting the formula (7), the formula (9) and the formula (13) into the above formula, the calculation can be obtained:
Figure SMS_54
wherein:
Figure SMS_55
Figure SMS_56
Figure SMS_57
Figure SMS_58
from the expression c 3 >0，c 4 >0, using dimensionless method, let:
Figure SMS_59
then equation (14) ultimately reduces to:
Figure SMS_60
order the
Figure SMS_61
μ=εδ+fcos (ωτ) then the system equation (17) equivalent system is:
Figure SMS_62
equation (18) is an undisturbed Hamilton system when ε=0, i.e
Figure SMS_63
It has three singular points: two saddle points (+ -1, 0) and a center (0, 0).
The Hamilton quantity of system formula (19) is:
Figure SMS_64
the trajectory into saddle point (0, 0) should be h when t tends to plus or minus infinity 0 =0, i.e.:
Figure SMS_65
two sink tracks with respect to time can be obtained by the above method:
Figure SMS_66
the above formula is expressed by integration and by an elliptic function:
Figure SMS_67
wherein: dn, sn and cn are Jacobi elliptic functions, q is an elliptic modulus. Closed rail period:
Figure SMS_68
wherein: k (q) is a first type of elliptic integral; t (T) q The elliptical mode q is monotonically increasing. />
Figure SMS_69
And->
Figure SMS_70
And->
Figure SMS_71
The Melnikov function for the cognate track is:
Figure SMS_72
calculated by
Figure SMS_73
When (when)
Figure SMS_74
When established, the Melnikov function M (t 0 ) There is a simple zero point. Therefore, under the above conditions, the system has chaos in the sense of a Smale horseshoe.
The parametric methods of the present invention are more clearly explained below in conjunction with a specific example. Assuming the thin wall plate is a titanium alloy TA15 material, its density ρ=4507 kg/m 3 Elastic modulus e= 102.04GPa, poisson ratio v=0.3, material linear expansion coefficient alpha 0 ＝8.4×10 -6 and/C. The thin wall plates have the length and width of a=b=1×10 -2 m, the initial displacement and velocity of vibration are x (0) =0m,
Figure SMS_75
substituting the above parameters into equation (24), and programming and calculating by four-order R-K method, wherein the tolerance error of integration is 10 -6 And obtaining a bifurcation diagram of system vibration through the programming calculation.
The invention analyzes the influence of the rotation speed of the main shaft, the cutting depth, the thickness of the thin-wall part and other factors on the vibration bifurcation and chaos of the milling process of the thin-wall part. In milling experiments, different cutting depths and spindle speeds are changed, and the intervals of the spindle speeds are set to be 1000rpm and the cutting depth a p The interval of the thin-wall parts is 0.05mm, and the thin-wall parts are milled for three times with the same parameters; also changing spindle speed and thin wallThe thickness of the thin-wall workpiece is set to be 1000rpm, the interval of the spindle rotating speed is set to be 0.05mm, the thin-wall workpiece is milled three times with the same parameters, the table 1 is a spindle rotating speed-cutting depth-vibration acceleration data table for milling the first thin-wall workpiece, and the table 2 is a spindle rotating speed-cutting depth-vibration acceleration data table for milling the second thin-wall workpiece. Table 3 is a spindle speed-Bao Biban thickness-vibration acceleration data table for the first thin-walled workpiece milling, and table 4 is a spindle speed-Bao Biban thickness-vibration acceleration data table for the second thin-walled workpiece milling. And drawing a relation chart of the rotating speed of the main shaft, the cutting depth and the vibration acceleration and a relation chart of the rotating speed of the main shaft, the thickness of the thin-wall part and the vibration acceleration.
Table 1 first spindle speed-depth of cut-vibration acceleration experiment
Figure SMS_76
Table 2 second spindle speed-depth of cut-vibration acceleration experiment
Figure SMS_77
TABLE 3 first spindle speed-Bao Biban thickness-vibration acceleration experiment
Figure SMS_78
TABLE 4 second spindle speed-Bao Biban thickness-vibration acceleration experiment
Figure SMS_79
According to the specific embodiment provided by the invention, the invention discloses the following technical effects:
(1) The invention adopts the algorithm of bifurcation and chaos theory, obtains the influence of milling parameters on the vibration of the nonlinear system through modeling the milling force of the thin-wall part, and stably predicts the nonlinear system.
(2) The invention can be applied to the selection of parameters in the real milling process by comparing and verifying the accuracy of the experiment and the calculated parameters.
In the present specification, each embodiment is described in a progressive manner, and each embodiment is mainly described in a different point from other embodiments, and identical and similar parts between the embodiments are all enough to refer to each other. For the system disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the relevant points refer to the description of the method section.
The principles and embodiments of the present invention have been described herein with reference to specific examples, the description of which is intended only to assist in understanding the methods of the present invention and the core ideas thereof; also, it is within the scope of the present invention to be modified by those of ordinary skill in the art in light of the present teachings. In view of the foregoing, this description should not be construed as limiting the invention.

Claims (4)

1. A vibration control method based on bifurcation and chaos analysis is characterized in that:
a dynamic milling force model in a modal direction is established;
a milling dynamics equation of the rectangular thin-wall piece under the action of milling heat is established according to the elastic mechanics of the plate shell,
the method comprises the following steps:
Figure QLYQS_1
wherein w is the dynamic displacement of the system in the mode direction, v is poisson ratio, ρ is the density of the rectangular thin-wall plate, h is the thickness of the rectangular thin-wall plate, and M T Is the heat moment, N T Is the heat power, the heat power is the heat power,
Figure QLYQS_2
as a stress function->
Figure QLYQS_3
For flexural rigidity, E is the elastic modulus;
t is time, a p For milling depth, w (t) is dynamic displacement of the system in the mode direction, and the included angle between w (t) and the x-axis is alpha, a 0 ＝a 11 cos 2 α+a 12 sinαcosα-a 21 sinαcosα-a 22 sin 2 α，a ij I, j=1, 2; t is the rotation period of each tooth, t=2pi/(nω), N is the number of teeth of the milling cutter; omega is the angular velocity of the spindle;
Figure QLYQS_4
Figure QLYQS_5
the expression of the thermal force and thermal moment is:
Figure QLYQS_6
alpha in the formula 0 Is the linear expansion coefficient of the material; t (x, y, z) is a temperature function generated by cutting forces during milling;
ω c derived from the following formulas (3) and (4):
Figure QLYQS_7
considering the case where w (t) is a constant amplitude harmonic, it is possible to obtain:
Figure QLYQS_8
wherein F is x,y Is to project the resultant force to the mode direction, and the cutter tooth j is positioned at the angle phi j Milling resultant force of Fx and Fy; fx, fy are two-degree-of-freedom bullets perpendicular to each otherA sexual damping system;
the chaotic vibration condition of the nonlinear dynamics system under Smale horseshoe transformation is judged based on a Melnikov function method;
solving a bifurcation diagram of the nonlinear system of the thin-wall part under the action of milling force;
the influence rules of cutting depth, cutting temperature, thin-wall part thickness, main shaft rotation speed and the like on the vibration of the nonlinear system are quantitatively analyzed, and the nonlinear system is stably predicted.
2. The method according to claim 1, characterized in that the model direction dynamic milling force model is in particular:
simplifying the degree of freedom of the milling system and projecting resultant forces to the modal direction;
and establishing an arbitrary time lag critical stable value of the system under the condition of taking the harmonic waves into consideration.
3. The method according to claim 2, characterized in that the milling force modeling is updated to thin-walled workpiece milling force Fang Guocheng modeling, in particular:
establishing a balance equation of a thin-wall rectangular plate with force and temperature coupling;
dynamic displacement in the modal direction of the thermal, force coupling is established.
4. A method according to claim 3, characterized in that the dynamic chaos and bifurcation decisions applied to the thin-wall plate milling process based on thin-wall plate milling force modeling are in particular:
solving an equation established by a Galerkin principle by using a dimensionless coefficient;
programming calculation by using a fourth-order R-K method, and setting the integral tolerance error to be 10 -6 And obtaining a bifurcation diagram of system vibration through programming calculation.
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